Overview
- Features a self-contained elaboration of the spectral theory for a specific integrable system
- Provides detailed proofs of all necessary asymptotic estimates
- Includes a complete treatment of singular spectral curves
Part of the book series: Lecture Notes in Mathematics (LNM, volume 2229)
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About this book
This book develops a spectral theory for the integrable system of 2-dimensional, simply periodic, complex-valued solutions u of the sinh-Gordon equation. Such solutions (if real-valued) correspond to certain constant mean curvature surfaces in Euclidean 3-space. Spectral data for such solutions are defined (following ideas of Hitchin and Bobenko) and the space of spectral data is described by an asymptotic characterization. Using methods of asymptotic estimates, the inverse problem for the spectral data is solved along a line, i.e. the solution u is reconstructed on a line from the spectral data. Finally, a Jacobi variety and Abel map for the spectral curve are constructed and used to describe the change of the spectral data under translation of the solution u. The book's primary audience will be research mathematicians interested in the theory of infinite-dimensional integrable systems, or in the geometry of constant mean curvature surfaces.
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Keywords
Table of contents (21 chapters)
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The Jacobi Variety of the Spectral Curve
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Front Matter
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Back Matter
Reviews
“The book is useful for specialists studying periodic solutions to integrable nonlinear partial differential equations.” (Dmitry E. Pelinovsky, Mathematical Reviews, October, 2019)
Authors and Affiliations
Bibliographic Information
Book Title: A Spectral Theory for Simply Periodic Solutions of the Sinh-Gordon Equation
Authors: Sebastian Klein
Series Title: Lecture Notes in Mathematics
DOI: https://doi.org/10.1007/978-3-030-01276-2
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer Nature Switzerland AG 2018
Softcover ISBN: 978-3-030-01275-5Published: 06 December 2018
eBook ISBN: 978-3-030-01276-2Published: 05 December 2018
Series ISSN: 0075-8434
Series E-ISSN: 1617-9692
Edition Number: 1
Number of Pages: VIII, 334
Number of Illustrations: 7 b/w illustrations
Topics: Partial Differential Equations, Differential Geometry, Functional Analysis, Functions of a Complex Variable, Ordinary Differential Equations